Completing the Square: GCSE Maths Step-by-Step

What does "completing the square" mean?

The phrase refers to creating a perfect square inside the expression. The square (x + p)² expands to x² + 2px + p². If you add p² to x² + bx, you get a perfect square — but you must also subtract it to keep the expression balanced. "Completing" means adding the missing piece to make the square exact.

The result, (x + p)² + q, is called the completed square form or vertex form.

How do you complete the square when a = 1?

For x² + bx + c:

Step 1 — Halve the coefficient of x: p = b ÷ 2.
Step 2 — Write (x + p)².
Step 3 — Subtract p² and add back c: the full expression is (x + p)² − p² + c.
Step 4 — Simplify the constant: q = c − p².

Worked example 1: complete the square for x² + 6x + 2

Step 1 — p = 6 ÷ 2 = 3.
Step 2 — (x + 3)².
Step 3 — (x + 3)² − 9 + 2.
Step 4 — (x + 3)² − 7.

Answer: x² + 6x + 2 = (x + 3)² − 7

Worked example 2: complete the square for x² − 8x + 5

Step 1 — p = −8 ÷ 2 = −4.
Step 2 — (x − 4)².
Step 3 — (x − 4)² − 16 + 5.
Step 4 — (x − 4)² − 11.

Answer: x² − 8x + 5 = (x − 4)² − 11

How do you complete the square when a ≠ 1?

When a ≠ 1, first factor out a from the x² and x terms.

Worked example 3: complete the square for 2x² + 12x + 7

Step 1 — Factor out 2 from the first two terms: 2(x² + 6x) + 7.
Step 2 — Complete the square inside the bracket: x² + 6x → (x + 3)² − 9.
Step 3 — Substitute back: 2[(x + 3)² − 9] + 7.
Step 4 — Expand the factor of 2: 2(x + 3)² − 18 + 7.
Step 5 — Simplify: 2(x + 3)² − 11.

Answer: 2x² + 12x + 7 = 2(x + 3)² − 11

How do you find the vertex of a parabola using completed square form?

The graph of y = (x + p)² + q is a parabola with its vertex (turning point) at (−p, q).

• If a > 0, the parabola opens upward and the vertex is a minimum point.
• If a < 0, the parabola opens downward and the vertex is a maximum point.

From Worked example 1: y = (x + 3)² − 7 has its minimum at (−3, −7).
From Worked example 2: y = (x − 4)² − 11 has its minimum at (4, −11).

The vertex x-coordinate is always −p (opposite sign to what is inside the bracket). This is the most common mistake: students write +3 instead of −3 for (x + 3)².

How do you solve a quadratic by completing the square?

Set the completed square form equal to zero and rearrange.

Worked example 4: solve x² + 6x + 2 = 0

From example 1: (x + 3)² − 7 = 0
(x + 3)² = 7
x + 3 = ±√7
x = −3 + √7 or x = −3 − √7

Solutions: x = −3 + √7 ≈ −0.35 or x = −3 − √7 ≈ −5.65

These solutions are left in surd form unless a decimal is asked for.

Summary of the completing the square process

Frequently asked questions

Why do we subtract p² after writing (x + p)²?

When you write (x + p)², you are secretly adding p² that was not in the original expression. To keep the expression equivalent, you must take p² away again. The subtract-p² step is the balancing act that keeps both sides equal.

Does completing the square always work?

Yes. Unlike factorising, completing the square works for any quadratic, even those with non-integer or irrational roots. It is the method used to derive the quadratic formula itself — so it is the most powerful of the three GCSE methods.

What is the line of symmetry of a quadratic graph?

The line of symmetry passes through the vertex and is vertical. Its equation is x = −p (the x-coordinate of the vertex). For y = (x + 3)² − 7, the line of symmetry is x = −3.

When should I use completing the square rather than the quadratic formula?

Use completing the square when a question asks you to find the minimum or maximum point of a curve, write an expression in vertex form, or derive exact surd solutions. Use the quadratic formula when you just want the numerical solutions and the question does not specify a method.

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